One future direction is to relax the distributional assumption on \(({\bf x},y)\) , say \({\bf x}\)  follows non-Gaussian distribution. Although there is a wide range of data may follow Gaussian assumption, e.g. abalone data and other biological data, many other data may not follow Gaussian, e.g. image data. The constant \(c_0\)  in our framework currently depends on the Gaussian assumption, and there is potential to relax it. Another one is concerned with sparse adversarially robust learning, say sparse \(\widehat{\theta }\) , which could be useful in both compressing and robustifying deep neural networks [1]}. The first step is to understand how the sparsity of \(\theta _0\)  (together with other model assumptions) implies that of \(\theta ^*\) , which in turn determines the sparsity of \(\widehat{\theta }\) . An example can be found in [2]} for linear sparse coding model. However, more careful studies would be needed in the future.
